Concerning the Design of Capacitively Driven Induction Coil Guns

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IEEE TRANSACTIONS ON PLASMA SCIENCE. VOL. 17. NO. 3. JUNF 19x9 429 Concerning the Design of Capacitively Driven Induction Coil Guns

Abstract-This paper is concerned with the design of capacitively 11. THEORYAND SIMULATIONMODEL driven, multisection, electromagnetic coil launchers, or coil guns, tak- A. Capacitively Driven Coil-huncher System ing their transient behavior into account. A computer simulation based on the viewpoint of lumped parameter circuits is developed to predict Fig. 1 is a sketch of a capactively driven, multisection the performance of the launcher system. It is shown that a traveling coil launcher which consists of two main parts: The barrel electromagnetic wave can be generated on the barrel by the resonance and the projectile. It may be driven either by a set of ca- of drive coils and their capacitors. More than half of the energy ini- pacitors, as indicated in the diagram, or, alternatively, by tially stored in the capacitor bank can be converted into kinetic energy a set of ac polyphase sources. The barrel is a linear array of the projectile in one shot, and an additional quarter can be utilized in subsequent shots, if the launcher dimensions, resonant frequency, of drive coils. The projectile is a conducting sleeve sur- and firing sequence are properly selected. The projectile starts smoothly rounding the payload. The drive coils on the barrel are from zero initial velocity and with zero initial sleeve current. Section- energized with a certain time sequence so as to generate to-section transitions which have significant effects on the launcher a traveling electromagnetic wave that interacts with a sys- performance are also discussed. Experimental results were obtained tem of currents in the sleeve. with a small model and are in good agreement with theoretical predictions. Two modes of operation are possible for the launcher: The “synchronous” mode and the “asynchronous” mode. In the first case, the velocity of the projectile is I. INTRODUCTION exactly equal to that of the traveling wave. For this mode a set of currents must be impressed on the sleeve; each N advantage that electromagnetic launchers have over coil must be energized separately; the timing of the clo- A chemical ones is that they can accelerate projectiles sure of each switch must be governed by the position of to much higher (hyper) velocities. The present interest in the projectile; and the size of each capacitor must be cho- electromagnetic launchers has resulted in a great compe- sen in such a way that the frequency matches the dM/dx tition between the two major types: Rail guns and coil curves of the drive coils. Because of the complexity of guns. Although rail guns are currently at a more advanced the control system and the power conditioner associated stage of development, coil guns have their own desirable with this mode, we discuss instead in detail here the asyn- characteristics which have been discussed in the literature chronous mode in which the projectile velocity is smaller [1]-[5]. The operating principles of various kinds of coil than that of the traveling wave so that the relative motion launchers have been summarized by Mongeau et al. [2]. between the two causes induced currents in the projectile. The performance of such launchers is still being investi- In this case the coils may be connected in series or in gated. In a previous paper [4], design principles based on parallel to form a limited number of phase windings, as the assumption that a steady state prevails in each section is done in conventional machines. However, since the of the gun were outlined. Here, instead, account is taken losses in the sleeve are proportional to its “slip” behind of the fact that the steady state is actually never reached the traveling wave, and since, in a launcher, the projectile during the transit of the projectile through the gun. velocity ranges over at least one order of magnitude, it The first part of this paper describes the physical ar- becomes necessary to increase the velocity of the travel- rangement and the computer simulation model. The sec- ing wave stepwise by dividing the barrel into sections. In ond part discusses the design. Numerical results and per- order to obtain a wave whose phase velocity increases formance estimates for a launcher now under construction from section to section along the length of the barrel, are given in the third part, which contains also a compar- either the frequency for the pole pitch (half wavelength) ison between theoretical predictions and experimental data must increase. Because of the relatively short length of obtained with a small laboratory model. the projectile it is not practical to increase the pitch ade- quately. Therefore the resonant frequency must increase Manuscript received October 14, 1988; revised February 8, 1989. This along the length of the barrel and different capacitor sizes work was supported by the SDIO/IST and managed by the AFOSR under must be used in different sections. Contract no. F49620-86-C-0126 and by the USASDC under Contract no. DASG60-88-C-0047. B. System Equations The authors are with the Polytechnic University, 333 Jay Street, Brook- lyn, NY 11201. The coil launcher of Fig. 1 is modeled by the equivalent IEEE Log Number 8927934. circuit shown in Fig. 2. Because the axial distribution of

0093-3813/89/0600-0429$01.OO 0 1989 IEEE 430 IEEE TRANSACTIONS ON PLASMA SCIENCE. Vol.. 17. NO. 3. JUNE IYXY

where [ C 3 is a diagonal (n element) matrix containing the capacitances and [ V,] and [Id ] are the capacitor voltage and the drive-coil current column matrices, respectively. [ V,] and [Id ] are submatrices of [ VI and [I3. The Lorentz force acting on the projectile is given by

F = 1 [I]TIG][I] (3)

where [ G 3 = d [ M ] /dx and the superscript T stands for Fig. l. Sketch of multisection capacitively driven coil launcher. the transpose of the matrix. Finally, we consider the equation of motion and combine it with (1)-(3). The complete set of equations which describe the capacitively driven coil-launcher system is then given in matrix form:

d { [L] + [MI} ;[I1 = [VI - [RI[I] - vp[G][I] (m + n eqns.) (4) d V IVcl = -[Id] (n eqns.) (5) + [cl Fig. 2. Lumped parameter circuit model of the capacitively driven coil launcher. the induced current in the sleeve is not uniform, to sim- ulate the physical system we divide the sleeve into several dx - = up (1 eqn.) zones that are electrically insulated from each other and dt represent them as separate projectile coils. The number of coils into which we divide the sleeve depends on the re- where Mp is the mass of projectile and up is the velocity quirement of accuracy. The number of drive coils de- of the projectile. Equations (4)-(7) represent a set of pends on performance specifications such as the muzzle simultaneous nonlinear differential equations with time- velocity, weight of the projectile, the caliber, and the variable coefficients. The number of first-order differen- length of the barrel. We denote by n the number of drive tial equations involved in the system is coils, and by m the number of projectile coils. Employing Kirchhoff’s voltage law, we write the net- N = 2n + m + 2 (8) work equations in matrix form: in 2n + m + 2 variables (mprojectile currents, n drive currents, n driver capacitor voltages, up, and x). [VI = [Rl[Il + { [LI[Il + [MI[Il) (1) f C. Analysis of the System Energies where [ VI and [I]are column (m + n element) matrices To assess the performance of the launcher system one composed of the individual projectile and drive-coil volt- needs first to consider the energy balance relations. This ages and currents, respectively. [ R ] and [ L] are diagonal will eventually lead to the optimum design of the launcher. (m + n element) matrices composed of the individual Basically, the energy balance involves several items: The projectile and drive-coil resistances and self-inductances, electric energy stored in the capacitor bank, the magnetic respectively, and [MIis a square (m + n) X (m + n) energy stored in the coils, the kinetic energy gained by matrix, each element of which represents a mutual induc- the projectile, and the ohmic loss due to the resistances. tance between any two individual coils. The formulas used The main goal in the design of a capacitively driven coil to evaluate each element of [ L]and [ M ] are given in [6] launcher is to achieve the transfer of as much as possible and [7]. It should be noted that the mutual inductances of the electric energy stored in the capacitors into kinetic [ M ] between drive coils and projectile coils are functions energy of the projectile to attain high system efficiency. of the distance x which denotes the position of the projec- The energy relations are derived as follows below. tile and is a function of time. This complicates matters Multiplying (1) by [I] and rewriting it in an alterna- considerably. tive form, we obtain: The relations between the capacitor voltages and drive- coil currents are HE et 01.: CONCERNING THE DESIGN OF CAPACITIVELY DRIVEN INDUCTION COIL GUNS 43 1

Since This average acceleration, as calculated by the simulation code, is an important quantity for the design of the launcher because it affects the dimensions of the launcher. Another key quantity is the Energy Transfer Ratio (ETR), d IT which is defined as the ratio of the gain in kinetic energy = [IIT([L] + [MI) [I1 -k 5 {[I] ~P[GI[I]) to the total energy initially stored in the capacitors, because it affects the weight of the capacitor bank:

where wkf and wk, refer to the final and initial kinetic energies of the projectile respectively, Wcap(0) stands for initial capacity energy, uj and U, are the final and initial We note in (1 1) that the term on the left represents the velocities of the projectile, and vcd(0)is the initial ca- electric power in the capacitor. The first term on the right- pacitor voltage. Finally, an Energy Loss Ratio (ELR) is hand side represents the ohmic loss of the system, the sec- defined as ond term represents the time rate of change of the magnetic energy stored in the coils, and the last term represents the converted power. Then the energy balance at time t can be derived from the power balance equation by Separating the ohmic losses of the projectile coils and the integrating (11) in time to obtain: drive coils and using (14), one obtains

Wcap = Wohm + Wmag + wconv (12) Wohm = w:hm + wfhm (21 1 where TP m w$hm = R,,]; dt (22) "1 S0 p=l Wcap = i: [IIT[V] dt' + c - C,Vf(O) (13) 1=l2 Td n wth,,, = c dt (23 1 S0 d=l Wohm = l: [I]T[R 1 [I] dt' (14) where Td is the energization (oscillation) time of a single section. This time may differ from one section to another. Using (21)-(23), one can write (20) as

TP ELR = n . (24) WcOnv= [I]'U~[G][I] dt' = - [IIT[G][I] dx 20s 20s' d= 1 CdV:d(O) ( 16 ) The temperature rise in the launcher can be derived from where 1 is the length of the barrel and Tp stands for the (22) and (23). If VvOl,pis the volume of a projectile coil, total time the projectile needs to travel along the barrel. vvol,d the volume of a drive coil, ~(JK-lrn-~)the specific The prime notation is used to distinguish between the van- heat Of the material, and e,, ed the temperature rise of able of integration t' and the upper limit t. each projectile coil and drive coil, respectively, and if one We observe from the integrands in (16) that the force assumes that the launch time is so short that the process on the projectile is is nearly adiabatic, then one obtains mn 1 dMdp F = - [IIT[G][I] = c I I (17) 2 p=l d=l dpdx which is the result we have used in the previous section. Since the system is time varying, we define an average acceleration mn JO 6d = d = 1, n. (26) dx cvvoI, d dt. If one assumes that the volumes of all the projectile MP (I8) coils are the same, then the average temperature rise in 432 IEEE TRANSACTIONS ON PLASMA SCIENCE. VOL. 17. NO 3. JUNE IYXY the projectile sleeve is ation, consistent with the maximum allowable mechanical and thermal stress. Then the kinetic energy gain WkJ in each drive-coil section is proportional to the length of the section, and may be expressed as

wkJ = Mpa,V 'J (29) Likewise, if one assumes that all the drive coils have where the subscript j stands for the jth section, 1, stands the same volume and that the time Td (that each drive coil for the length of thejth section, and Wk, is defined as is energized) is the same, then one can define the average 2 temperature rise of the drive coils as WkJ = iMp(v,2 -

n r\ T,J so that (30)

111. DESIGNOF A CAPACITIVELYDRIVEN COIL LAUNCHER Design Considerations The major considerations in designing a capacitively driven coil launcher are to determine the launcher dimen- where U = uj - vj - I is the velocity increment per section, sions, capacitor sizes, and the firing sequence needed from a quantity which is a system specification. vi- I, vj stands the power condition so that a traveling magnetic wave can for the entering and exit velocities of thejth section, re- be generated and a maximum energy transfer ratio can be spectively. The total length of the barrel is the summation obtained, while the mechanical and thermal stresses are of the section lengths: kept within the allowable limits for a given material. Unlike the case of most classical electric machines, in N, 1= c1, (33) the capacitively driven multisection coil launcher the j= 1 traveling wave is generated by the resonance of the drive coils and their capacitors. In addition, there are some pe- where N, is the number of sections. culiar problems. The traveling wave is a wave packet with It should be noted that the average acceleration aavused finite length and attenuation in time. This creates many in the above equations is selected on the basis of the max- time and space harmonics. The attenuation strongly de- imum allowable mechanical stress and represents an ini- pends on the coupling between the barrel and sleeve and tial input to the computer before the simulation is carried on their time constant. Good coupling gives good energy out. The energies stored initially in the capacitors of each transfer but high attenuation. The phase currents are section are then adjusted during the simulation so as to asymmetrical due to the asymmetrical mutual inductances yield corrected average accelerations from (18) which between the phases. This causes energy transfer between match those of (29) and (32). neighboring drive coils to occur and a different Capacitor Finally, it is advisable that the length of a section may Discharge Ratio (CDR) (the ratio of final-to-initial capac- be limited to a few wave lengths because of the attenua- itor energy storage) to exist in each phase. Besides, the tion of currents (see Section IV, below). resonant frequencies and the phase shifts in each phase 2) Drive Coils: The dimensions of the drive coils winding vary with the time and the position of the projec- should be chosen so that their radial thickness is the min- tile sleeve; in fact, the natural behavior of the coupled imum allowed by the mechanical and thermal stress limits oscillators tends to eliminate the phase shifts between the in order to have a good coupling with the sleeve. Their phase currents and thus kills the traveling wave on the axial width should also be optimized in order to reduce barrel. To design high-performance capacitively driven the energy transfer between the neighboring drive coils, coil launchers one should take all of the above problems while keeping the amplitude of the space harmonics within into consideration. First, for given dimensions one can an acceptable range. Due to consideration of the wave calculate the energy transfer ratio, the average accelera- attenuation, the resistances of the drive coils and of their tion, and the velocity gain in each section of the barrel. connecting leads should be as small as possible. This may These, in turn, lead to the determination of the quantities be achieved by reducing the number of turns per coil so which follow below. as to decrease the space needed for the insulation; how- I) Determination of Barrel Length: To estimate the to- ever, a low number of turns per coil (and hence a low tal length of a multisection barrel one can assume that the inductance) increases the capacitance and the current to average acceleration is constant for every section. This be handled in the power conditioner. assumption is based on the following consideration: For 3) Capacitances: As was mentioned before, each sec- the best utilization of the barrel one must maximize the tion consists of phase windings having the same fre- force acting on the projectile and, therefore, its acceler- quency. A first choice for the value of the phase capacitor HE er al.: CONCERNING THE DESIGN OF CAPACITIVELY DRIVEN INDUCTION COIL GUNS 433 is obtained from the steady-state relationship: IV. NUMERICALRESULTS AND DISCUSSION A. Highlights of Results 1 U1 J= -- (34) A capacitively driven four-section coil launcher which 2aa- 27 is currently being built is used as an example. The di- where4 is the frequency of the LC circuit in thejth section mensions of the launcher are given in Table I and the sim- and L,!,is the total self-inductance of the set of drive coils ulation results are given in Table 11. A four-section barrel which is connected to the capacitor. Since the equivalent with a total length of 1.8 m was designed to accelerate a inductance at the terminals of one phase is different from 127-g aluminum projectile from zero initial velocity to a its self-inductance due to the mutual inductance between muzzle velocity of 1000 m/s when the total initial energy phases, a coefficient K is used to account for this effect. stored in the capacitor bank is 122 kJ. The total ohmic C, is the capacitance per phase in thejth section and 7 is loss depends on the conductivity of the aluminum. With the pole-pitch of the machine. Hence the capacitance per some degree of pre-cooling, enough to increase the con- phase for thejth section is ductivity from y200cto 1.71 y200c,the total ohmic loss is about 24 percent and the kinetic energy gain in one shot (35) is about 52 percent, yielding a Capacitor Discharge Ratio (CDR) of 76 percent. The remaining 24 percent can be Since the steady state can never be reached, the values of utilized in subsequent shots. (If no pre-cooling is used, C, will be adjusted to produce the maximum ETR in (19) the ohmic loss is about 36 percent, with an ETR of 43 (see also Section IV, below). percent.) Details of the analysis are discussed in the fol- The volume of the capacitor bank required for a lowing paragraphs. launcher may be approximately estimated as follows: If one assumes that a 1-kg capacitor can store 1 kJ of energy B. Starting Section and that the specific weight of the capacitor is 4000 The starting section consists of six drive coils spanning kg/m3, the energy density of the capacitor bank is about one full wavelength (20 cm) and is connected in a three- 4 MJ/m3, a relatively small volume. phase configuration. Each phase is energized with a sep- 4) Voltage Levels: The ETR from (19) determines the arate capacitor. A total of 14.5 kJ is initially stored in the amount of energy which must be stored in the capacitor three-phase capacitors, with different voltages on each. in order to get a specified amount of kinetic energy. The This was necessary in order to achieve equal-amplitude voltage levels are then obtained from the following equa- currents in each of the phase windings. Otherwise the de- tions: lay in firing phases B and C and the asymmetrical mutual inductances between phases result in unequal current am- 1 wkj - C.V2 = - plitudes. The length of the aluminum sleeve equals the 2 ETR length of this section (or one wavelength); the thickness where Wkj is obtained from (30). The initial voltage on is 1.5 mm and the weight is 127 g. Initially, the left end each capacitor in thejth section is of the sleeve is positioned at the left end of the barrel. The dependence of the three-phase currents on time is shown in Fig. 3. Phase A is switched on at t = 0, then (37) 60" later phase C with a negative initial voltage, and then phase B follows 60" after phase C. The frequency and 5) Time Sequences for Switches: As mentioned in the phase-shift variations are caused by asymmetries in the previous section, the traveling wave on the barrel is re- mutual inductances between the phases due to the short- alized by energizing the drive coils in a certain time se- ness of the section. The waveshapes improve as the length quence. This requires that we predetermine the time se- of the section increases, as shown in Fig. 4 for the longer quence for every switch. last section (60 cm). According to (34), for an m-phase machine the time Current buildup in the sleeve can be observed in Fig. interval between two phases is 5. Fig. 5(a) shows the current distributions in the sleeve 27 at different instants of time: t = O', 0.033, 0.067, and t; - ti-1 = -. 0.1 ms with only phase A switched on (phases C and B muj will be switched on at t = 0.12 and 0.24 ms, respec- Hence, the ith phase in thejth section needs to be ener- tively). Comparison with Fig. 5(c) shows that at t = 0.1 gized at a time ti such that ms the sleeve current nearly approaches its peak, and this 27 gives a sleeve time constant of a fraction of a millisecond. t; = tj-1 + - The initial position of the sleeve is significant. Asym- m (39) vi metries in the induced current distribution along the sleeve In three-phase machines phase C may be fired at 1/6 of disappear, as shown in Fig. 5(b), when the sleeve is ini- a period (60" ) after A with a negative voltage, instead of tially placed 5 cm to the left of the position as compared at 1/3 of a period (120") later with a positive voltage. with Fig. 5(a) so that both drive coils in phase A are nearly 434 IEEETRANSACTIONS ON PLASMA SCIENCE, VOL. 17. NO 3. JUNE 1989

TABLE I three phase current IKAl 30.00 I DIMENSIONSOF AN 1 km/s FOUR-SECTIONLAUNCHER*

~~ Section one: Length 20 cm Number of coils 6 (15 turndcoil) Radial thickness of coil 1.5 cm Axial length of coil 3 cm Average Radius 3.425 cm Sections two, three and four have same cross section but different lengths and coils: Length (section 2.3.4) 40cm, 60cm, 60cm Number of coils (section 2.3,4) 12,18,18 ( 10 turndcoil each) Radial thickness of coil 1 cm Axial length of coil 3 cm \/ Average Radius 3.175 cm Sleeve: Length 20 cm Average radius 2.5 cm Radial thickness 1.5 mm Material aluminum Weight 127 g Clearance between sleeve and barrel coils 1 mm three phase current IKAl *The barrel consists of four sections with a total length of 1.8 m. The 40.00 coils in a section are connected in three phases with a pole pitch 7 of 10 cm.

TABLE I1 SUMMARYOF SIMULATION RESULTS

parameters section section section section #1 #2 #3 #4 results

stored caDacitor 14 25 41 41 5 122 energy (kJ) peak capacitor 4.5. 3.7 12. 9.3 24.9, 22.8 32.5. 29.4 voltage (kV) 3.3 7.6 21.1 27.1

peak current in 25 36'3 31'3 31'4 -40.08, I I I I I I I I I I I I 00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70' drive coilslkAi C Time Ins) Fig. 4. Currents in the three-phase wihdings in the last section.

energy initially stored in the capacitors has been dis- charged. The average temperature rise of the sleeve in the starting section is about 35°C. More information about the first section is given in Table 11.

I frePkUHez;CY (1 1 47 I 2 75 I 4.01 I 5.13 1) I C. Section-to-Section Transition A smooth transition from one section to the next can be obtained only when the phase shift between the traveling waves generated by the currents in the drive and projectile coils is maintained at a constant level. However, in a capacitively driven coil launcher, a traveling wave cannot equally spaced from the ends of the sleeve. The distri- be formed until all the three-phase windings of the section bution of the sleeve currents after all three-phase wind- are fully energized. For a brief interval of time during the ings have been energized is shown in Fig. 5(c). One can transition the leading section is single phased. This may see that a distorted traveling wave is moving from the left result in a change of sign in the accelerating force during to right. The distortion is caused by the space harmonics the transition and may lead to instability. It is possible, of the drive coils. Nevertheless, a relatively smooth ac- however, to alleviate the problem and to achieve a rela- celeration is achieved in the first section, as shown in Fig. tively smooth transition from one section to the next by 6. The projectile accelerates in the first section of the bar- carefully selecting the phase relations in the currents flow- rel from an initial velocity of zero up to 274.2 m/s. Of ing in the drive coils of the two sections. Fig. 7 shows the 14.5 kJ initially stored in the capacitors of this sec- the force and velocity profiles obtained after a few trials tion, at exit from the section 33 percent has been trans- for the four-section launcher. The force on the sleeve is ferred into kinetic energy, and 41 percent has been dis- relatively smooth within a section, but the transition in- sipated as ohmic loss. This means that 74 percent of the troduces a slightly negative force. HE et al. : CONCERNING THE DESIGN OF CAPACITIVELY DRIVEN INDUCTION COIL GUNS 435

current distribution lMA/ml force 110.2 KNI, velocity Ids1 10.00, I 350.00 8.00 300 .OO velocity

250.00

200.00 0.00 -2.00 \ -4.00 150'00100.00 \ t / \ 50.00 \ -10.00 ' // 0.00 \/,- -12.00 , "'/P/ -5O.O#, I 1 I I I I I I I -l4''# 00 2.00 4.00 6.00 8.00 10.00 12.00 14.00 16.00 18.00 20.00 00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.30 1 61eeve length lcml Time lmsl Fig. 6. Force and velocity profiles in the starting section.

Current distribut ion IMA/ml rn nn. velocity lm/sl. torcelxO.2 KNI 1000.00 900.00 - 800.00 - 700.00 -

600.00 - - 500.00 - 400.00 -

-6.00 ',\ /; -8.00 '. -4.001 I I I I -lO.O#, ?',iI -100.00 'I 00 2.00 4.00 6.00 E.00 10.00 12.00 14.00 16.00 18.00 20.00 1' ileeve length lcml , 00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00 3.25 3 (b) Tlme lms)

current distrlbution lHA/ml Fig. 7. Force and velocity profiles in the four-section launcher. 10.00

8.00

6.00 fusion is usually longer than the transit time of the projectile, the cumulative temperature rise in the sleeve is 4.00 nonuniform. An end-effect may also cause the tempera- 2.00 ture to peak at the two ends of the sleeve, as shown in 0.00 Fig. 8. One way to reduce this temperature-peaking effect -2.00 would be to increase the thickness of the sleeve at the two -4.00 ends, and in so doing add mechanical strength. The de- -6.00 pendence of total ohmic loss in the first section and tem- -8.00 - perature rise in the sleeve on the thickness of the sleeve

-10.08, I I I I I I I I I is shown in Fig. 9. The ELR is rather insensitive to the 00 2.00 4.00 6.00 8.00 10.00 12.00 14.00 16.00 18.00 20.00 ileeve length lcml sleeve thickness, but the temperature rise in the sleeve is (C) strongly dependent on its thickness. The sleeve thickness Fig. 5. (a) Buildup of sleeve current in the starting section. Only phase A should be chosen so that the ETR is maximized while the is energized. Left end of sleeve coincides with left end of barrel. (b) Buildup of sleeve current in the starting section. Only phase A is ener- temperature rise is kept within allowable limits. gized, but sleeve is initially placed 5 cm to the left. (c) Buildup of sleeve current in the starting section. All three-phase windings have been energized. Sleeve is moved 5 cm to the right. E. Skin Efect in the Sleeve In the previous discussions it was tacitly assumed that D. Ohmic Loss and Temperature Rise the radial current distribution in the sleeve was uniform; that is, that skin depth was not a significant factor. An Ohmic losses occur both in the drive coils and sleeve. estimate of the depth of penetration 6 may be obtained Simulation has shown that the ohmic loss in the drive coils from does not cause a serious temperature rise problem because of the short operating time. In the sleeve it is a different matter. Since the current induced in the sleeve is not dis- . (40a) tributed uniformly and the time constant of thermal dif- j&= j&= j& 436 IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 17. NO 3, JUNE 1989

temperature distribution i"c) 350 In CDR /--- '.. --- \ .\ 70 ', - .. - -. capacitance 60

ELR __/--- 30 ___-----

sleeve length IcmI 10 4 30 Fig. 8. Distribution of the cumulative average temperature rise in the sleeve. P.20 2.\0 2.160 frequency2.L (kHz1 3.\0 3.!0 3.\0 3!60° Fig. 10. Energy transfer ratio (ETR), energy loss ratio (ELR), capacitor discharge ratio (CDR), and capacitance versus the resonant frequency for the second section. Stored capacitor energy is 25 kJ. TABLE 111 DEPENDENCEOF SKIN DEPTH ON FREQUENCYFOR SLEEVEOF AN 1 km/s GUN

parameters section 1 section 2 section 3 section 4 frequency (Hz) 1470 2150 4010 51 30 sync. velocity (ds) 292 553 802 10% entering slip 1 0.5 0.31 0.13 depth of penetration > 1.86 1.92 2.02 2.28 (mm)

F. Energy Transfer From Capacitors to the Projectile

-- As we mentioned before, the major objective in design- 8.00i& 0.50 1.00 i.M 2.00 2.50 3.00 3.54 sleeve thickness inn1 ing a capacitively driven coil launcher is to convert as Fig. 9. Energy transfer ratio (ETR), energy loss ratio (ELR), capacitor much as possible of the energy stored in the capacitors discharge ratio (CDR), and sleeve temperature rise versus sleeve thick- into kinetic energy of the projectile. A low-energy con- ness for starting section. Stored capacitor energy is 14.5 kJ. Frequency is about 1750 Hz. version ratio may result from one of the following: 1) Failure to produce a proper traveling magnetic wave on the barrel. For po = 4a X H/m, y = 5 X lo7 mho/m, 7 = 2) Poor coupling between drive coils and the sleeve. 0.1 m, 6 becomes 3) Improper resonant frequency impressed on the section. 4) Improper sleeve dimensions and material conductivity. The dependence of the ETR on the resonant frequency The expression for 6 shows that the'depth of penetration of the second section is shown in Fig. 10, where one can in the sleeve depends on the slip frequency sus,and hence see that the ETR varies from about 25 to 59 percent as the on the slip velocity sus. For the best usage of the sleeve resonant frequency changes from 2.3 to 3.5 kHz. The material the current distribution in the radial direction ETR also strongly depends on the sleeve thickness, as should be relatively uniform, and therefore the simulation shown in Fig. 9. A thin sleeve gives a large resistance model used in this paper should not lead to a significant and high temperature rise while a thick sleeve yields a error. A relatively uniform current distribution is achieved long time constant and slow current buildup. Both ex- when the sleeve thickness is small as compared with the tremes result in a poor energy transfer ratio, as shown in skin depth 6. It follows that the frequency in each section Fig. 9. should be always chosen so that the slip is sufficiently small in order to satisfy the condition stated above. An- V. COMPARISONBETWEEN THEORETICAL PREDICTIONS other reason for selecting a small slip is that the heat loss AND EXPERIMENTALRESULTS is proportional to s. For the 1 km/s example (see below) we used here, the A small one-section demonstration model of a capaci- minimum depth of penetration in the 1.5-mm-thick sleeve tively driven coil launcher was built. Its parameters are occurs at the time when the sleeve enters each section as given in Table IV. Measured and calculated voltage and shown in the data in Table 111. Thus in each section, even current waves are shown in Figs. 11 and 12. Agreement at entry, the minimum skin depth exceeds the sleeve between the two is seen to be very good. Agreement be- thickness and the effect of radial nonuniformity is proba- tween the measured and calculated muzzle velocities, bly small. given in Table IV, is also very good. HE er al.: CONCERNING THE DESIGN OF CAPACITIVELY DRIVEN INDUCTION COIL GUNS 437

Results Total Average CapPh Init. Peak Peak Curr Transit Sleeve in VolWh perPh. Design Basis Time Temp. 3rd Sec. 3rd Sec. 3rd Sec. Transient 3.42 ms 128.7OC 51.8 pF 22.8 kV 31.3 kA (see Table 2) Steady-State I 371 ms I 143.6OC 1 82.8 P I (see ref. [41)

It should also be noted that the steady-state design procedure requires that several preliminary assumptions be made, such as ETR, average acceleration, and equivalent Fig. 11. Oscillogram of three-phase voltages and one-phase current for phase inductance. These must be based on experience and small laboratory model. Refer to Fig. 12. (Scales: 100 V, 50 A, 1 insight into the launcher operation [4]. Although the ms/div .) steady-state design procedure is much simpler, the transient approach provides a more detailed and accurate re- va VO Vc (VI 6 IC (A1x05 sult. VII. CONCLUDINGREMARKS 300 00 A lumped-parameter computer model of the coil 200 00 launcher gave performance predictions which agree with the experimental data obtained with a small-scale proto- 100 00 type that was built in the laboratory for proof-of-princi- ple. The model is a useful design tool, specifically with regard to the dimensioning of the coils and sleeve, and the sizing of section lengths and phase capacitors. It also -200 00 helps in establishing the proper phasing-in of successive

-300 00 sections so that the transition is relatively smooth. Al- though a complete design has been performed only for the -400 00‘ ’ I 0.00 1.00 2.00 3.00 4.00 5.00 6.00 700 8.00 1 km/s gun now under construction, preliminary calcu- Time Imsl lations suggest that muzzle velocities in the range of 6 Fig. 12. Computer simulation of three-phase voltages and one-phase current for small laboratory model. Refer to Fig. l l. km/s could be achieved with a relatively minor development effort. However, at higher velocities it is expected that a major effort to develop new types of switching ele- TABLE IV SMALLLABORATORY MODEL (REFER TO FIGS. 11 AND 12) ments will be required. ACKNOWLEDGMENT Barrel consists of one section with a total length of 20 cm. and the coils are connected m three phase with a pole pitch of 6.75 cm’ The authors would like to thank Prof. Y. Naot, H. K. Numher of coils 6 (100 turdcoil) Jung, H. P. Lee, P. Joshi, X. N. Lu, Q. C. Gu, and G. Radial thickness of coil 0.75 cm Axial length of coil 2.25 cm H. Fan for their contributions to the experimental work Coil inner diameter 6.7 cm and for their valuable discussions. Sleeve dimensions, Length 13.5 cm REFERENCES Radial thickness 0.17 cm [I] M. D. Driga, W. F. Weldon, and H. H. Woodson, “Electromagnetic Outer diameter 6 2 cm induction launchers,” IEEE Trans. Mugn., vol. MAG-22, pp. 1453- Material aluminum 1458, Nov. 1986. Weight llRg [2] P. Mongeau, W. Snow, G. Colello, P. Rezza, and H. Kolm, “Coilgun Capacitors used as power supply: technology evolution,” presented at the 4th Symp. on Electromagnetic Capacitance 288 PFiphase Launcher Tech., Austin, TX, Apr. 12-14, 1988. Initial Capacitor voltage 400 Viphase 131 D. G. Elliot, “Traveling-wave inductive launchers,” presented at the Muzzle vclocity (calculated) 3.5 ds 4th Symp. on Electromagnetic Launcher Tech.. Austin, TX. Apr. 12- Muzzle velocity (measured) 3.46 ds 14, 1988. -, IEEE Trans. Map., vol. 25, pp. 159-163. Jan. 1989. [4] Z. Zabar. Y. Naot, L. Birenbaum, E. Levi. and P. N. Joshi, “Design and power conditioning for the coil-gun.” presented at the 4th Symp. VI. COMPARISONBETWEEN TRANSIENT AND STEADY- on Electromagnetic Launcher Tech., Austin, TX, Apr. 12-14, 1988. STATE DESIGNAPPROACHES ____~-, and -, IEEE Trans. Mugn., vol. 25, pp. 627-631, Jan. 1989. [SI T. J. Burgess, E. C. Cnare, W. L. Oberkampf. S. G. Beard, and M. It is of interest to compare two designs for the same Cowan, “The electromagnetic 0 gun and tubular projectiles,” IEEE launcher: One based on the transient approach, as pre- Trans. Mag”., vol. MAG-18. pp. 46-59, Jan. 1982. sented in this paper, and the other on a steady-state ap- 16) F. W. Grover, Inductance Calculations. New York: Dover, 1973. 171 T. H. Fawzi and P. E. Burke, “The accurate computation of self and proach, presented earlier [4].Table V provides a few re- mutual inductance of circular coils,” IEEE Trans. Power App. SJst.. sults for comparison. vol. PAS-97, pp. 464-468, Mar./Apr. 1978. 438 IEEE TRANSACTIONS ON PLASMA SCIENCE. VOL. 17, NO. 3. JUNE 1989

Jianliang He (S’89) was born in Jiangsu, China, J. Hirsch Award, and the Lady Davis Fellowship, and is a member of Tau on December 27, 1954. He graduated from the Beta Pi and Sigma Xi. Department of Electrical Engineering, Xian Jia- otung University, Xian, China, in 1977. He * worked at the same University from 1977 to 1981. He received the M.S. degree from Rensselaer Zivan Zabar (M’76-SM’81) was born in Hadera, Polytechnic University, Troy, NY, in 1985. Since Israel, in 1939. He received the B.Sc., M.Sc.,and January 1985 he has been working towards ob- Sc.D. degrees from the Technion-Israel Institute taining the Ph.D. degree in energy conversion at of Technology in 1965, 1968, and 1972, respec- the Polytechnic University, Brooklyn, NY, where tively. currently he is a Research Associate. He is currently Professor of Electrical Engi- neering at the Polytechnic University in Brook- * lyn, NY. His areas of interest are power electron- ics and electrical power systems. Enrico Levi (M’58-SM’58-LS’89) is a graduate Dr. Zabar holds two patents and has published of the Technion-Israel Institute of Technology, many papers in technical journals. He is a member and received the Ph.D. degree in electrical engi- of Sigma Xi. neering from the Polytechnic University, Brook- lyn, NY. * He is Professor Emeritus of Electrophysics at the Polytechnic University, and is President and Leo Birenbaum (S’45-A’48-M’SS-SM’70) was Treasurer of Enrico Levi, Inc. He has been a Vis- born in New York City in 1927. He received the iting Professor and has lectured in many univers- B.E.E. degree from the Cooper Union, New York, ities in the United States and abroad, and has in 1946, the M.E.E. degree from the Polytechnic served as a Consultant to the U.S. DeDartment of Institute, Brooklyn, NY, in 1958, where he ob- Energy, NASA, the Argonne National Laboratory, the Lawrence Liver- tained the M.S. degree in physics in 1974. more Laboratory, and many industrial firms. He is now Research Associate Professor at the Dr. Levi’s published work includes: Electromechanical Power Conver- Polytechnic, where, since 1951, he has done re- sion (1982); Polyphase Motors: A Direct Approach to Their Design (1984); search and taught electrical engineering. He has chapters in Magnetohydrodynamic Electric Power Generation (1 978); worked and published in the areas of microwaves, Standard Handbook for Electrical Engineers (1986); The Encyclopedia of biological effects of magnetic fields, linear mo- Physical Science and Technology (1986); and more than 50 papers in tech- tors, and power distribution. nical and scientific journals. He also holds two patents. He is a recipient Mr. Birenbaum is a member of the New York Academy of Sciences and of the Sigma Xi Distinguished Faculty Research Citation, the IEEE Charles the Bioelectromagnetics Society.